# Square root, expressed as the limit of a sequence

Let \( c \in \mathbb{R} \) be a real. The sequence defined recursively below converges, and it converges to \( \sqrt{c} \).

Let \( x_1 = c \), and let \( x_n \) be defined like so:

Proof by induction (any other typical methods) can be used to show that sequences defined recursively like this do (or don't) converge.

### Intuition

The outer \( \frac{1}{2} \) factor can be seen as taking an average of the
two inner terms. Inspecting the second term, \( \frac{1}{2}(c + \frac{c}{c}) =
\frac{c + 1}{2} \), the third, \( \frac{1}{2}(\frac{c+1}{2} + c \frac{2}{c+1})
\), and so on, one can be convinced that this sequences is *increasing*
starting from the third term. In addition, it can be seen that
the terms don't surpass \( \sqrt{c} \). While not a exactly convincing as a
proof, you can check if a term *did* surpass \( \sqrt{c} \), then the
following term would be less than that term.